Fuzzy topology textbook pdf

FUZZY TOPOLOGICAL SPACES A THESIS SUBMITTED TO THE NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA IN THE PARTIAL FULFILMENT FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS BY TAPATI DAS UNDER THE SUPERVISION OF Dr. DIVYA SINGH DEPARTMENT OF MATHEMATICS NATIONAL INSTITUTE OF TECHNOLOGY, ROURKELA MAY, 2013

Abstract The present thesis consisting of three chapters is devoted to the study of Fuzzy topological spaces. After giving the fundamental definitions we have discussed the concepts of fuzzy continuity, fuzzy compactness, and separation axioms, that is, fuzzy Hausdorff space, fuzzy regular space, fuzzy normal space etc. 2

Acknowledgements I deem it a privilege and honor to have worked in association under Dr.Divya Singh, Assistant Professor in the Department of mathematics, National Institute of Technology, Rourkela. I express my deep sense of gratitude and indebtedness to him for guiding me throughout the project work. I thank all faculty members of the Department of Mathematics who have always inspired me to work hard and helped me to learn new concepts during our stay at NIT Rourkela. I would like to thanks my parents for their unconditional love and support. They have supported me in every situation. I am grateful for their support. Finally I would like to thank all my friends for their support and the great almighty to shower his blessing on us. 3

Contents 1. Preliminaries and Introduction 1.1 Fuzzy Set 1.2 Basic operation on Fuzzy Sets 1.3 Images and Preimages of Fuzzy Sets 2. Fuzzy topological space 2.1 Fuzzy topological space 2.2 Basis and Subbasis for FTS 2.3 Closure and interior of Fuzzy sets 2.4 Neighbourhood 2.5 Fuzzy Continuous maps 3. Compactness and Separation axioms 3.1 Compact fuzzy topological space 3.3 Fuzzy regular space 3.4 Fuzzy normal space 3.5 Other Separation axioms References 4

Chapter 1 Preliminaries and Introduction 1.1. Fuzzy Set Zadeh [11], 1965, introduced the concept of fuzzy sets by defining them in terms of mappings from a set into the unit interval on the real line. Fuzzy sets were introduced to provide means to describe situations mathematically which give rise to ill-defined classes, i.e., collections of objects for which there is no precise criteria for membership. Collections of this type have vague or fuzzy boundaries; there are objects for which it is impossible to determine whether or not they belong to the collection. The classical mathematical theories, by which certain types of certainty can be expressed, are the classical set theory and the probability theory. In terms of set theory, uncertainty is expressed by any given set of possible alternatives in situations where only one of the alternatives may actually happen. Uncertainty expressed in terms of sets of alternatives results from the nonspecificity inherent in each set. Probability theory expresses uncertainty in terms of a classical measure on subsets of a given set of alternatives. The set theory, introduced by Zadeh, presents the notion that membership in a given subset is a matter of degree rather than that of totally in or totally out. With fuzzy set theory, one obtains a logic in which statements may be true or false to different degrees rather than the bivalent situation of being true or false; consequently, certain laws of bivalent logic do not hold, e.g. the law of the excluded middle and the law of contradiction. This results in an enriched scientific methodology. Chang 1

[2], introduced the notion of a fuzzy topology of a set in 1968, and our work is based on the study of the properties of fuzzy topological spaces. Definition 1.1.1 [11]. Let X be a non-empty set. A fuzzy set A in X is characterized by its membership function µ A : X [0, 1] and µ A [x] is interpreted as the degree of membership of element x in fuzzy set A, for each x X. It is clear that A is completely determined by the set of tuples A = {[x, µ A [x]] : x X}. 1.2. Basic Operations on Fuzzy Sets Definition 1.2.1 [11]: Let A = {[x, µ A [x]] : x X} and B = {[x, µ B [x]] : x X} be two fuzzy sets in X. Then their union A B, intersection A B and complement A c are also fuzzy sets with the membership functions defined as follows: [i] µ A B [x] = max {µ A [x], µ B [x]}, x X. [ii] µ A B [x] = min {µ A [x], µ B [x]}, x X. [iii] µ A c[x] = 1 µ A [x], x X. Further, [a] A B iff µ A [x] µ B [x], x X. [b] A = B iff µ A [x] = µ B [x], x X. Lemma 1.2.2 [11]: The De Morgan s law are true for fuzzy sets. That is suppose A = {[x, µ A [x]] : x X} and B = {[x, µ B [x]] : x X} are fuzzy sets, then [A B] c = A c B c [1] [A B] c = A c B c [2] 2

Proof of equation [1]. We know that the following identity is true. 1 max[µ A, µ B ] = min[1 µ A, 1 µ B ] [3] To show that we consider the two possible cases: µ A µ B and µ A < µ B. If µ A µ B, then 1 µ A 1 µ B and 1 max[µ A, µ B ] = 1 µ A = min[1 µ A, 1 µ B ], which is equation [3]. If µ A < µ B, then 1 µ A > 1 µ B and 1 max[µ A, µ B ] = 1 µ B = min[1 µ A, 1 µ B ] which is again equation [3]. Hence this equation [3] is true. Now, the membership function of [A B] c is given by µ [A B] c[x] = 1 µ A B [x] = 1 max[µ A [x], µ B [x]] = min[1 µ A [x], 1 µ B [x]] = min[µ A c[x], µ B c[x]] = µ A c B c[x] This proves Equ. [1]. Similarly, using [3] we can prove Equ. [2]. 1.3 Images and Preimages of Fuzzy Sets Definition 1.3.1 [6]: The symbol I will denote the unit interval [0,1]. Let X be a nonempty set. Now, for the sake of simplicity of notation we will not differentiate between A and µ A. That is a fuzzy set A in X is a function with domain X and values in I, i.e. an element of I X. Let A, B I X and let f : X Y be a function. Then f[a] I Y, i.e. f[a] is a fuzzy set in Y, defined by 3

sup{a[x] : x f 1 [y]} if f 1 [y] φ f[a][y] = 0 if f 1 [y] = φ, and f 1 [B] is a fuzzy set in X, defined by f 1 [B][x] = B[f[x]], x X. Definition 1.3.2 [6]. The product f 1 f 2 : X 1 X 2 Y 1 Y 2 of mapping f 1 : X 1 Y 1 and f 2 : X 2 Y 2 is defined by [f 1 f 2 ][x 1, x 2 ] = [f 1 [x 1 ], f 2 [x 2 ]] for each [x 1, x 2 ] X 1 X 2. For a mapping f : X Y, the graph g : X X Y of f is defined by g[x] = [x, f[x]], for each x X. Definition 1.3.3 [6]. Let A I X and B I Y. Then by A B we denote the fuzzy set in X Y for which [A B][x, y] = min[a[x], B[y]], for every [x, y] X Y. Proposition 1.3.4 [6]. f 1 [B c ] = [f 1 [B]] c, for any fuzzy set B in Y. Proof. f 1 [B c ][x] = [B c ]f[x] = 1 B[f[x]] = 1 f 1 [B[x]] = [f 1 [B]] c [x], x X. Proposition 1.3.5 [6]. f[f 1 [B]] B, for any fuzzy set B in Y. Proof. The proof follows by noting that sup{f 1 [B][x] : x f 1 [y]}, if f 1 [y] φ f[f 1 [B][y]] = 0, if f 1 [y] = φ sup{b[f[x]] : x f 1 [y]}, if f 1 [y] φ = 0, if f 1 [y] = φ B[y], if f 1 [y] φ = 0, if f 1 [y] = φ Proposition 1.3.6 [6]. Let f : X Y be a mapping and A j be a family of fuzzy sets of Y, then 4

[a] f 1 [ A j ] = f 1 [A j ] [b] f 1 [ A j ] = f 1 [A j ] Proof-[a]. f 1 [ A j ][x] = [ A j ][f[x]] = [A 1 A 2 A j ][f[x]] = max{a 1 f[x], A 2 f[x],, A j f[x] } = max{f 1 [A 1 ][x], f 1 [A 2 ][x],, f 1 [A j ][x], } = [f 1 [A 1 ] f 1 [A 2 ] f 1 [A j ] ][x] = f 1 [A j ][x] [b]. f 1 [ A j ][x] = [ A j ][f[x]] = [A 1 A 2 A j ][f[x]] = min{a 1 f[x], A 2 f[x],, A j f[x] } = min{f 1 [A 1 ][x], f 1 [A 2 ][x],, f 1 [A j ][x], } = [f 1 [A 1 ] f 1 [A 2 ] f 1 [A j ] ][x] = f 1 [A j ][x] Proposition 1.3.7 [6]. If A is a fuzzy set of X and B is a fuzzy set of Y, then 1 [A B] = [A c 1] [1 B c ]. Proof. [1 [A B]][x, y] = max[1 A[x], 1 B[y]] = max[[a c 1][x, y], [1 B c ][x, y]] = [[A c 1] [1 B c ]][x, y] for each [x, y] X Y. 5

Proposition 1.3.8 [6]. Let f j : X j Y j be mappings and A j be fuzzy sets of Y j, j = 1, 2; then [f 1 f 2 ] 1 [A 1 A 2 ] = f 1 1 [A 1 ] f 1 2 [A 2 ]. Proof. For each [x 1, x 2 ] X 1 X 2, we have [f 1 f 2 ] 1 [A 1 A 2 ] = [A 1 A 2 ][f 1 [x 1 ], f 2 [x 2 ]] = min[a 1 f 1 [x 1 ], A 2 f 2 [x 2 ]] = min[f 1 1 [A 1 ][x 1 ], f 1 2 [A 2 ][x 2 ]] = [f 1 1 [A 1 ] f 1 2 [A 2 ]][x 1, x 2 ]. Proposition 1.3.9 [6]. Let g : X X Y be the graph of a mapping f : X Y. Let A be a fuzzy set of X and B be a fuzzy set of Y, then g 1 [A B] = A f 1 [B]. Proof. For each x X, we have g 1 [A B][x] = [A B]g[x] = [A B][x, f[x]] = min[a[x], B[f[x]]] = [A f 1 [B]][x]. 6

Chapter 2 Fuzzy Topological Space 2.1. Fuzzy Topological Space Definition 2.1.1 [6]. A family τ I X of fuzzy sets is called a fuzzy topology for X if it satisfies the following three axioms: [1] 0, 1 τ. [2] A, B τ A B τ. [3] [A j ] j J τ j J A j τ. The pair [X, τ] is called a fuzzy topological space or fts, for short. The elements of τ are called fuzzy open sets. A fuzzy set K is called fuzzy closed if K c τ. We denote by τ c the collection of all fuzzy closed sets in this fuzzy topological space. Obviously, we have: [a] α c τ c, [b] if K, M τ c, then K M τ c and [c] if {K j : j J} τ c, then {K j : j J} τ c. Example 2.1.2 [6]. Let X = {a, b}. Let A be a fuzzy set on X defined as A[a] = 0.5, A[b] = 0.4. The τ = {0, A, 1} is a fuzzy topology. [X, τ] is a fuzzy topological space. 0[a] = 0, a x, 1[a] = 1, a x. Let τ 1 and τ 2 be two fuzzy topology for X. If the inclusion relation τ 1 τ 2 holds, we say that τ 2 is finer than τ 1 and τ 1 is coarser than τ 2. 2.2 Base and Subbase for FTS 7

Definition 2.2.1 [1]. A base for a fuzzy topological space [X, τ] is a sub collection B of τ such that each member A of τ can be written as A = j Λ A j, where each A j B. Definition 2.2.2 [1]. A subbase for a fuzzy topological space[x, τ] is a subcollection S of τ such that the collection of infimum of finite subfamilies of S forms a base for [X, τ]. Definition 2.2.3. Let [X, τ] be an fts. Suppose A is any subset of X. Then [A, τ A ] is called a fuzzy subspace of [X, τ], where τ A = {B A : B τ}, B = {[x, µ B [x]] : x X} and B A = {[x, µ B A[x]] : x A}. Definition 2.2.4 [6]. A fuzzy point P in X is a special fuzzy set with membership function defined by λ if x = y, P [x] = 0 if x y; where 0 < λ 1. P is said to have support y, value λ and is denoted by P λ y or P [y, λ]. Let A be a fuzzy set in X, then P α y A α A[y]. In particular, P α y P β z y = z, α β. A fuzzy point P α y is said to be in A, denoted by P α y A α A[y]. The complement of the fuzzy point P λ x Definition 2.2.5 [6]. The fuzzy point P λ x is denoted either by P 1 λ x or by [P λ x ] c. is said to be contained in a fuzzy set A, or to belong to A, denoted by Px λ A if and only if λ < A[x]. Every fuzzy set A can be expressed as the union of all the fuzzy points which belong to A. That is, if A[x] is not zero for x X, then A[x] = sup{λ : Px λ, 0 < λ A[x]}. Definition 2.2.6 [6]. Two fuzzy sets A, B in X are said to be intersecting if and only if there exists a point x X such that [A B][x] 0. For such a case, we say that A and B intersect at x. 8

Let A, B I X. Then A = B if and only if P A P B for every fuzzy point P in X. Proposition 2.2.7 [6]. Let {A j : j J} be a family of fuzzy sets in X, P a x and P b y be fuzzy points in X and f be a map of X into Y. Then we have the following: 1. P a x {A j : j J} if and only if there exists j J such that P a x A j. 2. If P a x {A j : j J}, then for every j J we have P a x A j. 3. P a x P b y if and only if x = y and a < b. 4. If P a x P b y and for every j J, P b y A j, then P a x {A j : j J}. 5. If P a x A, where A is a fuzzy set in X, then there exists a < b such that P b x A. 6. f[p a x ] = P a f[x] 7. f[[p a x ] c ] = [f[p a x ]] c 8. If P a x A, then f[p a x ] f[a] 9. If Px a f 1 [B], then Pf[x] a B, where B is a fuzzy set in Y. 10. If P b y f[a], then there exists x X such that f[x] = y and P a x A. 11. If P b y B and y f[x], then for every x f 1 [y] we have P b x f 1 [B]. Proof. [1] P a x {A j : j J} if and only if there exists j J such that P a x A j. Let P a x A j a A j [x] a max{a j [x] : j J} a [ A j ][x]. Again P a x {A j : j J} a [ j J A j ][x] a A j [x] P a x A j, j J. 9

[4] If P a x P b y and for every j J, P b y A j, then P a x {A j : j J}. P a x P b y A j P a x A j a A j [x] a min{a j [x] : j J} a [ A j ][x] P a x {A j : j J}. [6] f[px a ] = Pf[x] a sup{p f[px a x a [z] : z f 1 [y]} if f 1 [y] φ ][y] = 0 if f 1 [y] = φ, = P a f[x] [y] y Y f[p a x ] = P a f[x] [7] f[[p a x ] c ] = [f[p a x ]] c a if x f 1 [y], = 0 otherwise; a if f[x] = y, = 0 otherwise; f[[p a x ] c ][y]= Now So 1 a if y = f[x], [Pf[x]] a c [y] = 0 otherwise; [i] 1 a if z = x, [Px a ] c [z] = 0 if z x; sup{[p f[[px a ] c x a ] c [z] : z f 1 [y]} if f 1 [y] φ ][y] = 0 if f 1 [y] = φ, 10

Thus f[[p a x ] c ] = [f[p a x ]] c. 1 a if x f 1 [y], = 0 otherwise; 1 a if f[x] = y, = 0 otherwise; [ii] Theorem 2.2.8 [8]. B is a base for an fts [X, τ] iff A τ and for every fuzzy point P in A, B B such that P B A. Proof. Assume that B is a base for τ, that is, every A τ is a union of members of B. Let A τ and P α x A. So A τ A = i I {B i : B i B} P α x A = i I {B i : B i B} P α x i I {B i : B i B} P α x B x A [for some B x ]. Conversely, assume that for each A τ and for each P α x A, B x such that P α x B x A. Let A τ. To prove that A can be written as a union of members of B consider any arbitrary P α x A. So by hypothesis B x B such that P α x B x A A P α x A B x. Since B x A, for each P α x A, therefore A = P α x A B. 2.3 Closure and Interior of fuzzy sets Definition 2.3.1 [6]. The closure A and the interior A o of a fuzzy set A of X are defined as A = inf{k : A K, K c τ} A o = sup{o : O A, O τ} respectively. 11

Example 2.3.2 [6]. Let A, B and C be fuzzy sets of I defined as 0 if 0 x 1 A[x] =, 2 2x 1 if 1 x 1; 2 1 if 0 x 1, 4 B[x] = 4x + 2 if 1 x 1, 4 2 0 if 1 x 1; 2 0 if 0 x 1 C[x] =, 4 4x 1 if 1 x 1; 3 4 Then τ = {0, A, B, A B, 1} is a fuzzy topology on I. It can be easily seen that Cl[A] = B c, Cl[B] = A c, Cl[A B] = 1, Int[A c ] = B, Int[B c ] = A and Int[A B] c = 0. 2.4 Neighborhood Definition 2.4.1 [6]. A fuzzy point P λ x is said to be quasi-coincident with A, denoted by P λ x qa, if and only if λ > A c [x], or λ + A[x] > 1. Proposition 2.4.2 [6]. Let f be a function from X to Y. Let P be a fuzzy point of X, A be a fuzzy set in X and B be a fuzzy set in Y. Then we have: 1 If f[p ]qb, then P qf 1 [B]. 2 If P qa, then f[p ]qf[a]. 3 P f 1 [B], if f[p ] B. 4 f[p ] f[a], if P A. 12

Proof. [1] Let P Px a, then sup{p f[px a x a [z] : z f 1 [y]} if f 1 [y] φ ][y] = 0 if f 1 [y] = φ; 0 if f 1 [y] = φ = a if x f 1 [y], iff[x] = y 0 if x / f 1 [y]; Now, f[p a x ] P a f[x] f[p a x ]qb = P a f[x] qb. Note that P a f[x] qb a +B[f[x]] > 1 a +f 1 B[x] > 1 P a x qf 1 [B], which completes the proof. Definition 2.4.3 [6]. A fuzzy set A in [X, τ] is called a neighborhood of fuzzy point P λ x if and only if there exists a B τ such that P λ x B A; a neighborhood A is said to be open if and only if A is open. The family consisting of all the neighborhoods of P λ x is called the system of neighborhoods of P λ x. Definition 2.4.4 [6]. A fuzzy set A in [X, τ] is called a Q-neighborhood of fuzzy point P λ x if and only if there exists a B τ such that P λ x qb A. The family consisting of all the Q-neighborhoods of P λ x is called the system of Q-neighborhoods of P λ x. Proposition 2.4.5 [6]. A B if and only if A and B c are not quasi-coincident; particularly, P λ x A if and only if P λ x is not quasi-coincident with A c. Proof. A[x] B[x] A[x] + B c [x] = A[x] + 1 B[x] 1. In particular P λ x A λ A[x] λ + A c [x] A[x] + A c [x] λ + A c [x] 1. Theorem-2.4.6 [6]. A fuzzy point e A o if and only if e has a neighborhood contained in A. 13

Theorem 2.4.7 [6]. A fuzzy point e = P λ x A if and only if each Q-neighborhood of e is quasi-coincident with A. Proof. P λ x A if and only if, for every closed set F A, P λ x F, or F [x] λ. By taking complement, this fact can be stated as follows: P λ x A if and only if, for every open set B A c, B[x] 1 λ. In other words, for every open set B satisfying B[x] > 1 λ, B is not contained in A c. From proposition, B is not contained in A c if and only if B is quasicoincident with [A c ] c = A. We have thus proved that Px λ A if and only if, for every open Q-neighborhood B of Px λ is quasi-coincident with A, which is evidently equivalent to what we want to prove. Definition 2.4.8 [6]. A fuzzy point e is called an adherence point of a fuzzy set A, if and only if, every Q-neighborhood of e is quasi-coincident with A. Theorem 2.4.9 [6]. [A] c = [A c ] o, [A c ] = [A o ] c. Definition 2.4.10 [6]. A fuzzy point e is called a boundary point of a fuzzy set A if and only if e A A c. The union of all the boundary points of A is called a boundary of A, denoted by b[a]. It is clear that b[a] = A A c. Definition 2.4.11 [6]. A fuzzy point e is called an accumulation point of a fuzzy set A if and only if e is an adherence point of A and every Q-neighborhood of e and A are quasi-coincident at some point different from supp[e], whenever e A. The union of all the accumulation points of A is called the derived set of A, denoted by A d. It is evident that A d A. Theorem 2.4.12 [6]. A = A A d, where A d is the derived set of A. Proof. Let Ω = {e : e is an adherence point of A}. Then, from Theorem [2.4.7] A = Ω. 14

On the other hand, e Ω is either e A or e / A ; from the Definition [2.4.11] we have e A d, hence A = Ω < A A d. The converse part follows directly. Corollary 2.4.13 [6]. A fuzzy set A is closed if and only if A contains all the accumulation points of A. Proof. We know that A = A A d. A fuzzy set A is closed if A = A and since A = A = A A d, therefore A d A. Conversely, if A contains all the accumulation point of A, then A d A and hence, A = A A d A = A. 2.5 Fuzzy continuous map Definition 2.5.1 [1]. Given fuzzy topological space [X, τ] and [Y, γ], a function f : X Y is fuzzy continuous if the inverse image under f of any open fuzzy set in Y is an open fuzzy set in X; that is if f 1 [ν] τ whenever ν γ. Proposition 2.5.2 [1]. [a] The identity id X : [X, τ] [X, τ] on a fuzzy topological space [X, τ] is fuzzy continuous. [b] A composition of fuzzy continuous functions is fuzzy continuous. Proof. [a] For ν τ, id 1 X [ν] = ν id X = ν. [b] Let f : [X, τ] [Y, γ] and g : [Y, γ] [Z, β] be fuzzy continuous. For η β, [g f] 1 [η] = η [g f] = [η g] f = f 1 [η g] = f 1 [g 1 [η]]. g 1 [η] γ since g is fuzzy continuous, and so [g f] 1 [η] = f 1 [g 1 [η]] τ since f is fuzzy continuous. Proposition 2.5.3 [1]. Let [X, τ] be fuzzy topological space. Then every constant function from [X, τ] into another fuzzy topological space is fuzzy continuous if and only if τ contains all constant fuzzy sets in X. 15

Proof. Suppose that every constant function from [X, τ] into any fuzzy topological space is fuzzy continuous and consider the fuzzy topology γ on [0, 1] defined by γ = { 0, 1, id [0,1] }. Let k be a real number, 0 k 1. The constant function f : X [0, 1] defined by f[x] = k, for every x X, is fuzzy continuous, and so f 1 [id [0,1] ] τ. But for x X, f 1 [id [0,1] ][x] = id [0,1] [f[x]] = id [0,1] [k] = k, whence the constant fuzzy set k in X belongs to τ. Conversely, suppose that τ contains all constant fuzzy sets in X and consider a constant function f : [X, τ] [Y, γ] defined by f[x] = y 0. If ν γ, then for any x X we have f 1 [ν][x] = ν[f[x]] = ν[y 0 ], so that f 1 [ν] is a constant fuzzy set in X and hence, a member of τ. Thus, f is fuzzy continuous. 16

Chapter 3 Compactness and Separation Axioms 3.1. Compact Fuzzy Topological Space Definition 3.1.1 [1]. A fuzzy topological space [X, τ] is compact if every cover of X by members of τ contains a finite subcover, i.e. if A i τ, for every i I, and i I A i = 1, then there are finitely many indices i 1, i 2,, i n I such that n j=1a ij = 1. Theorem 3.1.2. Let [X, τ] and [Y, γ] be fuzzy topological spaces with [X, τ] compact, and let f : X Y be a fuzzy continuous surjection. Then [Y, γ] is also compact. Proof. Let B i γ, for each i I, and assume that i I B i = 1 Y. For each x X, i I f 1 [B i ][x] = i I B i [f[x]] = 1 X. So the τ-open fuzzy sets f 1 [B i ][i I] cover X. Thus, for finitely many indices i 1, i 2,, i n I, n j=1f 1 [B ij ] = 1 X. If B is any fuzzy set in Y, the fact that f is a surjection mapping onto Y implies that, for any y Y, f[f 1 [B]][y] = sup{f 1 [B][z] : z f 1 [y]} = sup{[b]f[z] : f[z] = y} = B[y] f[f 1 [B]] = B. Thus, 1 Y = f[ 1 X ] = f[ n j=1[f 1 [B ij ]]] = n j=1f[f 1 [B ij ]] = n j=1b ij. Therefore, [Y, γ] is also compact. Lemma 3.1.3 [Alexander Subbase Lemma] [1]. If S is a subbase for a fuzzy topological space [X, τ], then [X, τ] is compact iff every cover of X by members of S has a finite sub cover [i.e. if A α S for each α Λ and α Λ A α = 1, then there are finitely many indices α i, [i = 1, 2,, n] such that n i=1a αi = 1]. Definition 3.1.4 [1]. Let [X i, τ i ] be a fuzzy topological space, for each index i I. The 17

product fuzzy topology τ = i I τ i on the set X = i I X i is the coarsest fuzzy topology on X making all the projection mappings π i : X X i fuzzy continuous. Theorem 3.1.5 [Fuzzy Tychonoff Theorem] [1]. Let n be a positive integer and for each i = 1, 2,, n, let [X i, τ i ] be a compact fuzzy topological space. Then [X, τ] = [ n i=1 X i, n i=1 τ i] is compact. Proof. We will say that a collection of open fuzzy sets of a fuzzy topological space has the finite union property [FUP] if none of its finite sub collections cover the space [i.e. none of its finite sub collections have supremum identically equal to 1]. Since S = {π 1 i [A i ] : A i τ i, i = 1, 2,, n} is a subbase for [X, τ], by the Lemma it suffices to show that no subcollection of S with FUP covers X. Let C be a sub collection of S with FUP. For each i = 1, 2,, n let C i = {A τ i : π 1 i [A] C}. Then C i is a collection of open fuzzy sets in [X i, τ i ] with FUP. Indeed, if A i,1, A i,2,, A i,k C i satisfy k j=1 A i,j = 1 Xi, then k j=1 π 1 i [A i,j ] = π 1 i [ k j=1 A i,j] = π 1 i [ 1 Xi ] = 1 X, and this would contradict the fact that C has FUP. Therefore, by the compactness of [X i, τ i ], the collection C i cannot cover X i, and we can select a point x i X i such that [ C i ][x i ] = a i < 1. Now if we consider the point x = [x 1, x 2,, x n ] X and the collection C i = {π 1 i [A] : A τ i } C, then it follows that [ C i][x] = {π 1 i π 1 i [A] C} = [ C i ][x i ] = a i. [A][x] : A τ i and π 1 i [A] C} = {A[x i ] : A τ i and Further noting that C = n i=1 C i, we obtain [ C][x] = n i=1 [ C i][x] = n i=1 [ C i ][x i ] = n i=1 a i which is strictly less than 1 since each of the finitely many real numbers a i is strictly less than 1. Thus C 1, as desired. 18

3.2. Fuzzy Regular Space Definition 3.2.1. An fts [X, τ] will be called regular if for each fuzzy point P and each fuzzy closed set C such that P C = 0 there exist fuzzy open sets U and V such that P U and C V. Proposition 3.2.2. Every subspace of regular space is also regular. Proof. Let X be a fuzzy regular space and A be a subspace of X. We have to prove that A is regular. Recall that τ A = {G A : G τ}, where G = {[x, µ G [x]] : x X} and G A = {[x, µ G A[x]] : x A}. Let P α x be fuzzy point in A and F A is closed set of A such that P α x / F A. Since A is a subspace of X, therefore P α x X and there is a closed set F in X, which generated the closed subset F A of A. Since X is regular space and P α x F = 0 there exist open sets U and V such that P α x U = [x, µ U ] and F V = [x, µ V ]. Thus U A = [x, µ U A ], V A = [x, µ V A ] are open sets in A such that P α x U A and F A V A. Hence A is a regular subspace of X. Proposition 3.2.3. If a space X is a regular space, then for any open set U and a fuzzy point P X such that P U = 0, there exists an open set V such that P V V U. Proof. Suppose that X is a fuzzy regular space. Let U = {[x, µ U ] : x X} be a fuzzy open set of X such that Px α U = 0. Then U = [x, 1 µ U ] is fuzzy closed set of X such that Px α / U = [x, 1 µ U ] and hence, Px α U. Since X is regular, therefore there exist two disjoint fuzzy open set V and W such that P α x V and U W. Now W is a closed set of X such that V W U. Thus, P α x V V and V W U and hence, V U. This proves that P α x V V U. 19

3.3. Fuzzy Normal Space Definition 3.3.1. A fuzzy topological space [X, τ] will be called normal if for each pair of fuzzy closed sets C 1 and C 2 such that C 1 C 2 = 0 there exist fuzzy open sets M 1 and M 2 such that C i M i [i = 1, 2] and M 1 M 2 = 0. Proposition-3.3.2. If a space X is a normal space, then for each closed set F of X and any open set G of X such that F G = 0 there exists an open set G F such that F G F G F G. Proof. Let X be a normal space. Let F be a fuzzy closed set in X and G be an fuzzy open set in X such that F G = 0, then F G. Let G = [x, µ G ] and F = [x, µ F ], then F and G are two disjoint fuzzy closed sets of X. Since X is fuzzy normal, so two disjoint fuzzy open sets G F and G G such that F G F and G G G and G F G G = 0. Thus, G F G G, but G G is a fuzzy closed set and hence G F G G. Thus from the above we have F G F G F G. 3.4. Other Separation Axioms Definition 3.4.1 [9]. An fts [X, τ] is said to be fuzzy T 0 iff x, y X, x y, U τ such that either U[x] = 1 and U[y] = 0 or U[y] = 1 and U[x] = 0. Definition 3.4.2 [a] [9]. An fts [X, τ] is said to be fuzzy T 1 - topological space iff x, y X, x y, U, V τ such that U[x] = 1, U[y] = 0 and V [y] = 1, V [x] = 0. Definition 3.4.2[b] [7]. An fts [X, τ] is F T 1 iff singletons are closed. Definition 3.4.3[a] [7]. An fts [X, τ] is said to be Hausdorff or fuzzy T 2 iff the following conditions hold: 20

If p, q are any two disjoint fuzzy points in X then [i] if x p x q, open sets V p and V q, such that p V p, q / V p, q V q, p / V q ; [ii] if x p = x q, and µ p [x p ] < µ q [x p ], then an open set V p such that p V p, but q / V p. Definition 3.4.3[b] [8]. A fts [X, τ] is said to be fuzzy Hausdorff iff for any two distinct fuzzy points p, q X, there exist disjoint U, V τ with p U and q V. Definition 3.4.4 [7]. An fts [X, τ] is F T 3 iff it is T 1, or F T 1 and regular. Definition 3.4.5 [7]. An fts [X, τ] is F T 4 iff it is T 1, or F T 1 and normal. Proposition 3.4.6. Every subspace of T 1 -space is T 1. Proof. Let X be a T 1 fuzzy topological space and A be a subspace of X. So τ A = {G A G A = [x, µ G A ], G τ}. Let x, y A such that x y. Then x, y X are two distinct points and as X is T 1, there exist U, V τ such that U[x] = 1, U[y] = 0 and V [y] = 1, V [x] = 0. Then, U A and V A are fuzzy open sets of A such that U A [x] = 1, U A [y] = 0 and V A [y] = 1, V A [x] = 0. This shows that A is also T 1. Theorem 3.4.7 [8]. A fuzzy subspace of a fuzzy Hausdorff topological space is fuzzy Hausdorff. Proof. Let X be a fuzzy Hausdorff topological space and A be a subspace of X. Let Px α, Py β be any two arbitrary points in A with Px α Py β. Then, we have Px α, Py β X, with Px α Py β. Since, X is a Hausdorff space therefore U, V τ such that Px α U, Py β V and U V = 0. Since U, V are fuzzy open subsets of X and µ U [z] µ V [z] = 0, for every z X, therefore U A = [x, µ U A ] and V A = [x, µ V A ] are fuzzy open subsets of A such that Px α U A, Py β V A and U A V A = 0,. Thus [A, τ A ] is also a fuzzy Hausdorff topological space. 21

Proposition 3.4.8 [7]. No subset of a Hausdorff fts can be compact. Corollary 3.4.9 [7]. Singletons in an Hausdorff fts are not compact. Theorem 3.4.10 [8]. If {[X i, τ i ] : i I} is a family of fuzzy Hausdorff topological spaces, their product[x, τ] is also fuzzy Hausdorff. Proof. Let {X i : i I} be a family of fuzzy Haudorff spaces and X = i I X i. We have to show that X is fuzzy Hausdorff. Let Px α, Py β X with Px α Py β. We know that the projection P i : X X i, i I is fuzzy continuous. Px α Py β there exists some i 0 I such that say Px α = m and P β y = n, P i0 [m] = P i0 [n] m i0 n i0 and we have P i0 : X X i0 and here m i0, n i0 X i0 with m i0 n i0. X i0 is fuzzy Hausdorff there exists open sets U and V in X i0 such that m i0 U and n i0 V and U V = φ. P 1 i 0 [U] open X and P 1 i 0 [V ] open X. Since P i0 is continuous m i0 U P i0 [m] U m P 1 i 0 [U] again n i0 V P i0 [n] V n P 1 i 0 [V ]. Claim. P 1 i 0 [U] P 1 i 0 some q P 1 i 0 [V ] = φ. Suppose to the contrary P 1 i 0 [U] P 1 i 0 [U] P 1 i 0 [V ] φ. This [V ] q P 1 [U] and q P 1 [V ] P i0 [q] U and i 0 P i0 [q] V q i0 U and q i0 V U V = φ which is a contradiction. Therefore [X, τ] is also fuzzy Hausdorff. Proposition 3.4.11. Every subspace of T 3 -space is T 3. Proof. We know that T 3 = T 1 +Regular. The proof follows by noting that every subspace of T 1 -space is T 1 and every subspace of regular space is regular. i 0 Proposition 3.4.12. Every subspace of T 4 -space is T 4 Proof. We know that T 4 = T 1 +Normal. Since every subspace of T 1 -space is T 1 and every 22

subspace of normal space is normal, therefore every subspace of a T 4 -space is T 4. Theorem 3.4.13 [7]. An F T 2 -space is an F T 1 -space. Proof. Let p be a fuzzy point in X. Then any point q {p} belongs to an open set V q such that µ {p} [x p ] µ V q [x p ]. So V q {p}. If, on the other hand, p is crisp, let x q X {x p } be arbitrary. If {q n, n N} be a sequence of fuzzy points, where x qn = x q, n N and the sequence {µ qn [x q ], n N} is decreasing and converges to zero, then there exists a sequence of open sets {V pqn, n N}, such that p V pqn and q n / V pqn, n N, as [X, τ] is Hausdorff. Therefore, if P = n N V pq n, then P is a closed set, where µ p [x q ] = 0 and µ p [x p ] = 1. So P is an open set contained in {p} and containing the crisp point q. Theorem 3.4.14 [7]. An F T 3 -space is an F T 2 -space. Proof. Let p, q be two fuzzy points, where x p x q and let w be a third fuzzy point, where x w = x p and µ w [x p ] > 1 µ p [x p ]. Then {w} is open and 1 µ w [x p ] < µ p [x p ] for x = x p, µ {w} [x] = 1 otherwise. Therefore q {w}, but p / {w}. Now since [X, T ] is regular, there exists V q τ such that q V q V q {w}. Obviously then, p / V q. Similarly, an open set V p can be determined such that p V p and q / V p. Theorem 3.4.15 [7]. An F T 4 -space is an F T 3 -space. Proof. Let [X, τ] be a regular space. Let p X and V τ. Since X is F T 4 it is F T 1 and normal. Since X is F T 1. {p} is a closed set in X. Since X is normal. There exists G τ such that {p} G G V p G G V. 23

References [1] S. Carlson, Fuzzy topological spaces, Part 1: Fuzzy sets and fuzzy topologies, Early ideas and obstacles, Rose-Hulman Institute of Technology. [2] C. L. Chang, Fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 24 [1968], 182 190. [3] R. Lowen, Fuzzy topological spaces and fuzzy compactness, Journal of Mathematical Analysis and Applications, 56 [1976], 621 633. [4] J. N. Mordeson, Zadehs influence on mathematics, Scientia Iranica D,18[3] [2011], 596-601. [5] J. R. Munkres, Topology, Prentice Hall Inc., New Jersey, 2000. [6] N. Palaniappan, Fuzzy topology, Narosa Publications, 2002. [7] M. Sarkar, On fuzzy topological spaces, Journal of Mathematical Analysis and Applications, 79 [1981], 384 394. [8] R. Srivastava, S. N. Lal and A. K. Srivastava, Fuzzy Hausdorff topological spaces, Journal of Mathematical Analysis and Applications, 81 [1981], 497 506. [9] R. Srivastava, S. N. Lal and A. K. Srivastava, On fuzzy T 0 and R 0 topological spaces, Journal of Mathematical Analysis and Applications, 136 [1988], 66 73. [10] S. Willard, General Topology, Dover Publications, New York, 2004. 24

[11] L. A. Zadeh, Fuzzy sets, Information and Control, 8 [1965], 338-353. 25